Damping of Wind Turbine Tower Oscillations through Rotor Speed ...

model with concentrated parameters. This can
be done using modal analysis that is very
common tool in wind turbine analysis [1], [3].
It describes a complex oscillatory structure as
a composition of several simple oscillatory
systems each of them being described by
means of mass, stiffness and damping. By this
representation complex tower oscillations are
seen as a sum of many simple oscillations
characterized by their modal frequencies
which are one of the most important structural
properties of wind turbine. It has been shown
in practice [5] that fairly good modeling of
wind turbine tower nodding can be achieved
using two modal frequencies (two modes).
Since we are here primarily interested in
building model suitable for controller design
we use only the first modal frequency. The
justification for this lies in the fact that for the
turbine in scope second modal frequency is
more than 6 times greater than the first modal
frequency and therefore falls out of the
controller frequency bandwidth.
By using only one modal frequency tower
dynamics can be described as:
Mx �� + Dx� + Cx = F , (9)
t t t
where M, D, and C are modal mass damping
and stiffness respectively and F is the
generalized force that is originated by wind
and that causes wind turbine tower
oscillations. Tower modal properties in
expression (9) are related to first tower modal
frequency ω0t as follows [4]:
D= 2 ζ ω ⋅M,
t 0t
2
0t
( ω )
C = ⋅M,
(10)
where ζ t is structural damping. For steel
structure structural damping is mostly set to
0.005 [4]. Modal mass M can be calculated as
[1]:
ht
0
( ) ( ) 2
φ
M = ∫ m h h dh,
(11)
where ht is the height of the tower, m(h) is the
mass distribution along the tower and φ ( h)
is
the tower's first mode shape. Note that actual
distribution of mass along tower has to be
modified in order to include mass of the rotor
and the nacelle which is assumed to be
concentrated at the tower top.
Driving force F is mostly the rotor thrust force
Ft caused by wind. It can be shown [4] that
thrust force, similar to aerodynamic torque,
depends upon wind speed, rotor speed and
pitch angle. So similarly to (6) it can be
expressed as [4]:
1 2 2
Ft = ρairR πvwCt( λ, β)
, (12)
2
where Ct is so called thrust coefficient.
Expressions (6), (8), (9) and (12) form the
simplified nonlinear model of wind turbine
that is used in the following sections for
controller design. Model is summarized below
taking into account the fact that wind speed
seen by the rotor is a sum of wind speed and
tower nodding speed:
J � tω= Mr −Mg −Ml,
1 3
2
Mr = ρairR πCQ( λ, β)
⋅( vw −x�t)
,
2
. (13)
1 2
2
Ft = ρairR πCt( λ, β)
⋅( vw −x�t)
,
2
F = Mx �� + Dx�+ Cx.
t t t t
Torque and thrust coefficients Cq and Ct are
usually provided by wind turbine blade
manufacturers or can be calculated using
professional simulation tools.
4. PID Controller
The PID controllers are still by far the most
used controllers for wind turbine speed and
power control. This is due to their simplicity
and rather high robustness. The small number of
parameters makes possible for designer to
quickly arrive at satisfactory, although in many
cases suboptimal, system behavior. As stated
before wind turbine dynamics change in
nonlinear fashion with change in wind speed.
To control the wind turbine with linear
controllers gain scheduling has to be used [3].
Although it seems straightforward to use wind
speed as scheduling criterion this is not an
appropriate solution since the wind speed is not
measured fast and accurately enough. Therefore
measured pitch angle is usually used as
scheduling variable.
For the design of a PID controller using
analytical methods process model (13) has to
be linearised around chosen operating point.
After linearization of expressions (13) and